1. Field of the Invention
The present invention relates in general to a design method for a semiconductor circuit, more specifically, to a design method for a semiconductor circuit characterized by a field-effect transistor having plural gate electrodes to extract characteristics of a circuit which the transistor is mounted on.
2. Background of the Related Arts
In development of semiconductor devices such as LSIs, circuit simulation is an important process to achieve a shortened development period by predicting characteristics of a circuit prior to prototype production. In a traditional standard circuit simulation, current-voltage characteristics of a transistor, a primary circuit-constituent element, have been described in use of a simple model which is not based on a physical model. BSIM (Berkeley Short-Channel IGFET Model) is one of typical examples thereof. As to this model, however, many expect that the number of device parameters required for accurate reproduction of circuit operations in simulation would have to increase every year to keep abreast with micronization of LSI and complication of processes, and it will also become more difficult to make model parameters coincide exactly with device parameters. Under these circumstances, a new circuit simulation model, which is built based on a transistor physical model represented by HiSIM (Hiroshima-University STARC IGFET Model), has recently been suggested. Current (I) flowing between source and drain of a transistor similar to the ones shown in FIG. 3 and FIG. 4 (hereinafter referred to as MOSFET, Metal Oxide Semiconductor Field Effect Transistor) in such a model in general is determined through multiplication of charge density by mobility of an inversion layer at a gate surface as follows:
                    I        =                              W            L                    ⁢                                    ∫              0                              V                ds                                      ⁢                                          μ                ⁡                                  (                  V                  )                                            ⁢                                                Q                  inv                                ⁡                                  (                  V                  )                                            ⁢                              ⅆ                V                                                                        (        1        )            where, L and W indicate length and width as shown in FIG. 4, and Vds indicates a voltage between source and drain. In Equation (1), mobility (μ(V)) is one of device parameters determining current characteristics of a circuit simulator, and is determined by the effect of scattering in electrons or holes inside an inversion layer.
Scattering mechanisms determining a value of the mobility can be classified according to their causes. Examples of major causes include an oscillation of channel-constituent atoms, interactions with channel impurities, and roughness at the gate surface, which are respectively called phonon scattering, Coulomb scattering, and (surface) roughness scattering.
Their contributions to the mobility follow Matthiessen's rule expressed as follows:1/μ=1/μph+1/μimp+1/μrs  (2)where, μ is measured mobility, and μph, μimp, and μrs are respectively mobilities provided that phonon scattering, Coulomb scattering, and roughness scattering are only dominant scattering mechanisms.
Under a certain temperature, phonon scattering maintains a constant level, but Coulomb scattering varies depending on the impurity density in a channel and the charge density of an inversion layer. Meanwhile, roughness scattering is caused by interactions between a gate surface and an inversion layer charge, and varies in magnitude (or level) according to individual manufacturing processes such as material of a gate oxide film, state of surface, etc. Therefore, when the roughness scattering is introduced as a device parameter to a circuit simulator, it is vital and indispensable to extract the roughness scattering dependency on devices by all kinds of manufacturing processes.
The following will now explain how to extract roughness scattering limited mobility.
The roughness scattering limited mobility is influenced by interactions between the inversion layer charge and the gate/oxide interface, so it varies depending on a distance between charge center of the inversion layer and a gate insulating film. This distance also varies by an electric field in a direction normal to the gate insulating film. Therefore, for a bulk MOSFET similar to the ones shown in FIGS. 3 and 4, the dependency of mobility on scattering mechanisms is evaluated by means of a drawing or a graph having intensities of a so-called effective normal electric field (Eeff) plotted along its axis. Here, the effective normal electric field can be expressed in terms of Qdep, Qinv, and ∈si as follows:Eeff=(ηQinv+Qdep)/∈si  (3)
where Qdep is a charge density of a depletion layer in a channel, Qinv is a charge density of an inversion layer, and ∈si is a dielectric constant of silicon.
Also, η is defined to ½ for an NMOSFET, and ⅓ for a PMOSFET.
Further, mobility may be evaluated by using a value of a linear area where Vds, which is called effective mobility
                              μ          eff                =                  LI                                    WQ              inv                        ⁢                          V              ds                                                          (        4        )            in Eq. (1), is proportional to a current vale.
Generally, an effective electric field has a value between about 0 MV/cm and 1 MV/cm. In a high electric field close to 1 MV/cm, the charge density in an inversion layer increases, and the center of charge in the inversion layer draws near to the gate surface. As Coulomb scattering gets weaker by electric shielding effects, roughness scattering becomes dominant compared to Coulomb scattering. In related to this, there has been a report in IEEE Transactions on Electron Devices, vol. 41, p 2357, 1994, for example, with regard to a bulk transistor of various channel impurities with different concentrations from each other, asserting that, under a certain temperature, a envelope is drawn if mobility is plotted as a function of Eeff, being overlapped in a high electric field.
Since this envelope has a fixed value being independent of gate electric thickness or impurity concentration of individual devices, it is called a universal mobility curve. Thus, a device parameter of roughness scattering is determined by extracting this curve.
Other terms like Qinv in Eq. (1) and Qdep in Eq. (3) for a bulk transistor are determined by equations explained, for example, in K. K. Schroder “Semiconductor Material and Device Characterization 2nd Edition” Wiley-Interscience Publication, John Wiley & Sons Inc, pp. 541 (1998), in which an inversion layer capacity Cinv and an accumulation layer capacity Cacc having been obtained through a split-CV method are substitutedQinv=∫−∞VCinvdV  (5)Qdep=∫VfbVCaccdV  (6)as follows:
where V indicates a gate voltage impressed to a transistor, and Vfb (called a flat band) is defined as a voltage at which the charge density of a channel becomes 0 (null).